The Shiny App I have produced compares the difference in using a linear model as opposed to a quadratic model when the data generated is quadratic.

The formula I used to generate the synthetic data was of form \(y = -0.5x^2 + 10x -18\) and in the range \(x = [3, 18]\). I also generated some gaussian noise to add to the y-term of the form \(y \sim \mathcal{N}(\mu =0, \sigma^2=1)\).

The code to generate this is down below:

x_values <- seq(3, 18, by = 0.1)
y_values <- -0.5 * x_values^2 + 10 * x_values - 18
set.seed(123)
noise <- rnorm(length(y_values), mean = 0, sd = 1)
y_noisy <- y_values + noise
data <- data.frame(x = x_values, y = y_noisy)

Here is a visualisation of the data generated:

In the Shiny app we then fit the two models to the data and do a visualisation of the fits as well as the models prediction for a chosen value of x (set by the user).

# Include x^2 term in dataframe for use in quadratic model
data$x2 <- (data$x)^2
fit_linear <- lm(y~x, data = data)
fit_quad <- lm(y~x+x2, data = data)

The residual plots all appears they should be for each of the diagnostic measures, showing the quadratic was a good fit!